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Step 1/2: Explorations in Mathematics and Science Teaching Title of Lesson: The Triangles are Congruent, Now What? UFTeach Students’ Names: Rachel Edwards and Jacob Lehenbauer Teaching Date and Time: October 25, 2012 10:30-12:45 Length of Lesson: 50 minutes Grade / Topic: 10-11th grade Honors Geometry Source of the Lesson: Jurgensen, Ray C., Richard G. Brown, and John W. Jurgensen. Geometry. Evanston, IL: McDougal Littell, 2000. Print. Cooperative Learning and High School Geometry: Becky Bride; Kagan Publishing; www.KaganOnline.com Concepts This concept follows from those learned by the students previously in proving triangles congruent. When triangles are introduced, the most useful ability, in relation to proving geometric theorems, is proving triangles congruent to each other. This can be done with various information, the minimum necessary information needed is three corresponding congruent pieces. These pieces can be all three sides congruent, two pairs of corresponding congruent angles with a pair of corresponding congruent sides in between, or two pairs of corresponding congruent sides with a pair of corresponding congruent angles in between. The theorems related, named Side Angle Side and so on, can be used to prove triangles congruent. After using these theorems, it is useful to know additional information that follows from these theorems. CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. This is useful for concluding much more about the related triangles and proving other things thereafter. It is important to remember positioning of the triangles, as these triangles may be oriented different directions. http://www.mathwarehouse.com/geometry/congruent_triangles/congruent-parts-CPCTC.php Warren Buck, J. Pahikkala. "CPCTC" (version 4). PlanetMath.org. Freely available at http://planetmath.org/CPCTC.html. Florida State Standards (NGSSS): MA.912.G.4.6: Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. Cognitive Complexity: Moderate Process Skills needed for this lesson: ● solving a problem requiring multiple operations ● retrieving information from a graph, table, or figure and using it to solve a problem ● providing a justification for steps in a solution process ● comparing figures or statements Performance Objectives ● Given two congruent triangles, students will be able to state which sides of the two triangles are corresponding to each other. ● Given two congruent triangles, students will be able to state which angles of the two triangles are corresponding to each other. Step 1/2: Explorations in Mathematics and Science Teaching ● Students will be able to prove that corresponding parts of congruent triangles are congruent by using either SSS, SAS, ASA, or AAS theorems. Materials List and Student Handouts ● Appendix A, x24 per class ● Appendix B, x24 per class ● Appendix C, x24 per class ● Appendix D, x 24 per class Advance Preparations ● Teachers will have a copy of the appendices they wish to go over with the class for viewing on the document camera ● Teachers will have all the worksheets ready to be handed out into groups Safety ● There are no significant safety concerns with this lesson. 5E Lesson Templates ENGAGEMENT Time: 5 minutes What the Teacher Will Do Teacher Directions and Student Responses and Probing Questions Potential Misconceptions The teacher will give Appendix A Today I want to start with to the students. going over what you have learned before. I want you to do this worksheet by yourself and we will go over the results. The teacher will encourage the students to exercise their prior knowledge of the methods of proving triangles congruent to review and give them confidence for the lesson. (give students a few minutes to complete the worksheet) Do we already know how to prove two triangles congruent? How? [Yes! By using the SAS, SSS, AAS and ASA theorems.] Can all triangles be proved Guide the students to the congruent using the same concept of CPCTC by asking for theorem? additional information about the [No, we need to use different theorems depending on what prior information we have.] Step 1/2: Explorations in Mathematics and Science Teaching triangles. Even if we have 3 congruent sides or angles in any order, is that enough to prove that triangles are congruent? [No, for example, problem 3 on the worksheet gives SSA. This is not enough information to prove that two triangles are congruent.] Is there anything else we can now conclude? How would [The students can have various we prove them? answers here, hopefully including the relationships of the other parts of the triangle.] EXPLORATION minutes What the Teacher Will Do Time: 15 Teacher Directions and Probing/Eliciting Questions This activity was a good review for what we are going to explore next. You all had great observations! This next worksheet will be done in groups. Talk with your group to reason out these problems together. Make sure you give correct justification for your reasoning. The teacher should How can we prove these two monitor the groups triangles congruent? Which progresses as they work, theorem can we use from asking guiding/probing before? questions as they do. Student Responses and Misconceptions Hand out Appendix B. Teacher will explain that we are looking to see if we can find the triangles congruent using things we already know. [Answers will vary, but they should use one of the SAS, SSS, etc. theorems.] Students may try to use other methods still, they should be guided to the use of the theorems they learned previously. Students cannot use AAA or SSA for theorems! [Shouldn’t those be congruent too?] Students should be led to the fact that all the other parts are also congruent. Step 1/2: Explorations in Mathematics and Science Teaching Now we have the triangles Students may think that they cannot congruent, good! Does that assume this, but it should be pointed mean we can say anything out that since the triangles are proven else about them? What about to be congruent, every single part of the other parts we didn’t use? the two triangles needs to be congruent to its corresponding part. EXPLANATION minutes What the Teacher Will Do Time: 15 Teacher Directions and Student Responses and Probing/Eliciting Questions Misconceptions Let’s look at problem 1. (Call on student) How did you conclude that triangle AED is [SAS (their reasoning will congruent to triangle BEC? vary) ] Why? The teacher will go over the results of Appendix B with the students, hearing how they proved the triangles congruent (to give practice with other material) and question them about other conclusions that can How did you conclude that be drawn about the triangles the measure of angle D is once they are proved congruent. equal to the measure of angle [Since the two triangles are C? (Call on another student) congruent, the corresponding angles should also be For problem 2, how did you congruent.] conclude that triangle PQO is congruent to triangle RSO? [ASA (their reasoning will From this, how did you vary) ] conclude that O is the midpoint of QR? (call on [since the triangles are another student) congruent, QO is congruent to RO. Thus O is the midpoint of QR by definition of midpoint.] For problem 3, how did you conclude that triangle JMO is congruent to triangle KMO? [AAS, their reasoning will vary)] (call on a student) From this, how did you conclude that M is the Step 1/2: Explorations in Mathematics and Science Teaching midpoint of JK? (call on another student) [since the triangles are congruent, JM is congruent to KM. Thus O is the midpoint of JK by definition of midpoint.] For problem 4, how did you conclude that triangle NOP is congruent to triangle SRP? [AAS (their reasoning will (call on a student) vary) ] From this, how did you conclude that NP is congruent to SP? (call on another student) [since both of the triangles are congruent, their sides that match up will be equal.] Now that we know these two triangles are congruent, is there anything else we can [Yes, their other parts look to see about the triangles? (call be the same too. Which makes on another student) sense, if the triangles are both congruent.] After going over these answers, does the fact that two triangles are congruent [Yes, because all of the give us additional information corresponding angles and the about the other angles and sides of the two triangles sides in the figures? should be equal/congruent.] What can we use this information to do? To use this, we just say the fact that “Corresponding parts of Congruent triangles are Congruent.” How is this new information useful? [We could conclude other things about the triangles, like the other corresponding angles and the other corresponding sides are congruent.] [This could be useful because we can use those congruences to help us prove other things about the figure!] Possible misconceptions Step 1/2: Explorations in Mathematics and Science Teaching using AAA or SSA to prove congruent triangles is incorrect. From this, we can see that two triangles are congruent if and only if their vertices can be matched up so that the corresponding parts of the triangles are congruent. So lets review steps to proving two corresponding segments or two corresponding angles are congruent (write these steps on the board) 1) Identify two triangles in which the two corresponding segments or corresponding angles are corresponding parts 2) Prove that the triangles are congruent 3) State that the two corresponding parts are congruent using the reason “Corresponding Parts of Congruent Triangles are Congruent” (CPCTC) EVALUATION minutes What the Teacher Will Do It should be made sure that the students know these things can only be concluded after the triangles have been proven congruent by one of the previously used theorems. Students could also see different parts of the triangles congruent based on how they are arranged. Positioning should be stressed, so students are always using the correct corresponding parts. Time: 15 Assessment Student Responses Step 1/2: Explorations in Mathematics and Science Teaching Hand out Appendix C (one per student). Tell the class that they will now be doing an evaluation. They are to do the work on their own. Step 1/2: Explorations in Mathematics and Science Teaching Appendix A Name: ____________________________________________________________________________ Directions: Determine whether the following triangles are congruent. If so, complete the congruence statement by filling in the blank and write the postulate or theorem that justifies the congruence. Otherwise, state that the “congruence cannot be proven.” E B D C A Justification: ∆𝐵𝐶𝐴 ≅ ∆______? M F A R K ∆𝐴𝐾𝑀 ≅ ∆______? Justification: O L M N P Step 1/2: Explorations in Mathematics and Science Teaching Justification: ∆𝐿𝑀𝑁 ≅ ∆______? Appendix B Name: ________________________________________________________________________________________ Using Congruent Triangles to Show Other Congruencies Directions: In your group, construct two-sided proofs for each of the figures below using the given statements. #1 #2 A B P Q E O C D Given : AE @ CE DE @ BE Pr ove : ÐA @ ÐC #3 R S Given : ÐP @ ÐS O is the midpoint of PS Pr ove : O is the midpoint of QR #4 Step 1/2: Explorations in Mathematics and Science Teaching O N S 1 O 3 1 2 4 K M 2 R P Given : NO ^ OR SR ^ OR Given : Ð1 @ Ð2 Ð3 @ Ð4 < 1 @< 2 Pr ove : M is the midpoint of JK Prove : NP @ SP Appendix C NO @ SR Step 1/2: Explorations in Mathematics and Science Teaching Name: _________________________________________________________________________________ Given: The two triangles below are congruent. D C O A B Fill in the blank and provide reasons: 3) ̅̅̅̅ 𝐴𝑂 ≅ ________? 1) ∆𝐴𝐵𝑂 ≅ ∆__________? ̅̅̅̅ ≅ ________? 4) 𝐵𝑂 2) ∠ 𝐴 ≅ ∠ ______? 5) Using what you have learned in the previous chapters and what you learned today to complete a twosided proof using the given statements below: Q 3 P 2 1 4 5 R S GIVEN: 4 5,QR SR PROVE: 2 3